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Mathematics LibreTexts

2.1: Lines

1. Lines (definitions)

Everyone knows what a line is, but providing a rigorous definition proves to be a challenge.

Definition: Line

A line with slope \(m\) through a point \(P = (a,b)\) is the set of all points \((x,y)\) such that

\[\dfrac{y-b}{x-a}= m.\]

2. The Slope Intercept Form of the equation of a Line

Given a point \((x_1,y_1)\) and a slope \(m\), the equation of the line is

Definition: Slope Intercept Equation of a Line

\[y-y_1=m(x-x_1)\]

3. Piecewise Linear Functions

A function is piecewise linear if it is made up of parts of lines

Example 1

\[f(x)=\begin{cases} x+4 & \text{if }x\leq-2 \\ 2x-1 & \text{if } -2<x<1 \\ -2x & \text{if } x\geq1\end{cases}\]

We graph this line by sketching the appropriate parts of each line on the same graph.

4. Applications

Example 2

Suppose you own a hotel that has 150 rooms.  At $80 per room, you have 140 rooms occupied and for every $5 increase in price you expect to have two additional vacancies.  Come up with an equation that gives rooms occupied as a function of price.

Solution

Let \(x\) be the price of a room and \(y\) be the number of rooms occupied.  Then we have an equation of a line that passes through the point \((80,140)\) and has slope \(-\frac{1}{5}\).  Hence the equation is:

\[y - 140 = -\dfrac{1}{5}(x - 80)\]

or 

\[y = -\dfrac{1}{5} x + 16 + 140\]

or

\[y = -\dfrac{1}{5} x + 156.\]

Exercise 1

What should you do if your two year old daughter has a 40 degree C temperature?  

Hint:  We have the two points:  \((0,32)\) and \((100,212)\).

Exercise 2

Suppose that your company earned $30,000 five years ago and $35,000 three years ago.  Assuming a linear growth model, how much will it earn this year?

Exercise 3

My rental was bought for $204,000 three years ago.  Depreciation is set so that the house depreciates linearly to zero in twenty years from the purchase of the house.  If I plan to sell the house in twelve years for $250,000 and capital gains taxes are 28% of the difference between the purchase price and the depreciated value, what will my taxes be?

Exercise 4

Wasabi restaurant must pay either a flat rate of $400 for rent or 5% of the revenue, whichever is larger.  Come up with the equation of the function that relates rent as a function of revenue

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